What is the practical utility of the central limit theorem in applied statistics?

The Central Limit Theorem (CLT) is one of the most important concepts in applied statistics, with widespread practical utility in a variety of fields. It is a fundamental result in probability theory that describes the behavior of the sum of a large number of independent, identically distributed random variables. The theorem states that as the sample size increases, the distribution of the sample means tends towards a normal distribution, regardless of the shape of the original population distribution. This has significant practical implications for many areas of applied statistics, including hypothesis testing, confidence intervals, and the analysis of survey data.

Hypothesis testing is a critical tool in statistics that allows us to determine whether a difference between groups or a relationship between variables is statistically significant. The central limit theorem provides a foundation for many common hypothesis tests, such as the t-test and the z-test. These ___ ____ _____ ______ ___.
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